Phase space partition with Koopman analysis.

Area of Science:

• Nonlinear dynamics
• Dynamical systems theory
• Computational physics

Background:

• Symbolic dynamics is crucial for analyzing nonlinear systems.
• Determining partition boundaries for symbolic dynamics is challenging.
• Existing methods often struggle with complex systems.

Purpose of the Study:

• To develop a robust method for symbolic partitioning in nonlinear systems.
• To overcome the challenge of defining partition boundaries.
• To apply the method to chaotic maps.

Main Methods:

• Constructing eigenfunctions of the finite-dimensional approximation of the Koopman operator.
• Identifying partition boundaries by locating the extrema of these eigenfunctions.
• Improving accuracy by increasing the number of basis functions in numerical computations.
• Validating the scheme on established 1D and 2D chaotic maps.

Main Results:

• A novel approach to symbolic partitioning for chaotic maps is presented.
• Partition boundaries are effectively determined by the extrema of Koopman operator eigenfunctions.
• The accuracy of the symbolic partition improves with a higher number of basis functions.
• The method is successfully demonstrated on benchmark 1D and 2D maps.

Conclusions:

• The proposed method provides an effective way to perform symbolic dynamics on chaotic maps.
• This technique simplifies the analysis of topological features in nonlinear systems.
• The Koopman operator eigenfunctions offer a promising avenue for advanced dynamical system analysis.